Some examples of this include: Uber/Lyft, decreasing transportation costs and improving quality for millions, while creating tens of thousands of jobs; Airbnb, same thing for hotels; social apps like Twitter, Instagram, etc, increasing entertainment/information flow to users, creating new media careers; etc. So startups don't need to operate any differently than they do now ... Think big and make big moniez!
One thing that definitely does not work are "social enterprise" startups. Distraction from the brutal realities of market forces ultimately leads these to fail while patting themselves on the back for doing "good."
In the spirit of the question though I'll list a few areas I think have enormous potential.
Money services and lending - very difficult to innovate here due to the iron fist of Uncle Sam and banking cartels. I think what's needed is either, 1) US-caliber teams operating outside the US or 2) products that skirt the law a la Uber/Airbnb but like those products are so positive with users that going against them becomes a risk for politicians.
Health - outsourcing/telepresence, reducing the cost of basic testing. Again may need to operate outside the US.
Legal - enormous value to be unlocked by standardizing and automating this field a la Clerky but better. Difficult to get lawyers on board, they are very good at extracting rents.
Real estate - breaking regulatory capture of Realtors.
End of life services - we're all in this market. It's not sexy so not much innovation.
Education - it's so backwards that you can innovate in any dimension.
Investment - despite nominal progress on things like crowdfunding, there really hasn't been a big impact yet by startups in increasing access to capital / opportunities to invest. It's still far too difficult to raise money for new businesses. Again, may have to operate outside of the US for this and consider innovative legal structures.
Science - breaking the University stranglehold on credible scientific research through crowdsourced/funded efforts. Enormous potential to cheaply fund studies especially those that go against orthodoxy or corporate interests. Again may need to be outside US.
